The random discrete action for two-dimensional spacetime

Dionigi M T Benincasa (1) , Fay Dowker (1) , Bernhard Schmitzer (1, 2)

1 Blackett Laboratory
2 Universität Heidelberg [Heidelberg] = Heidelberg University

A one-parameter family of random variables, called the Discrete Action, is defined for a 2-dimensional Lorentzian spacetime of finite volume. The single parameter is a discreteness scale. The expectation value of this Discrete Action is calculated for various regions of 2D Minkowski spacetime, M 2. When a causally convex region of M 2 is divided into subregions using null lines the mean of the Discrete Action is equal to the alternating sum of the numbers of vertices, edges and faces of the null tiling, up to corrections that tend to zero as the discreteness scale is taken to zero. This result is used to predict that the mean of the Discrete Action of the flat Lorentzian cylinder is zero up to corrections, which is verified. The " topological " character of the Discrete Action breaks down for causally convex regions of the flat trousers spacetime that contain the singularity and for non-causally convex rectangles.

Mots clés: Physical Sciences

Domaines: Relativité Générale et Cosmologie Quantique [gr-qc]

Dionigi M T Benincasa, Fay Dowker, Bernhard Schmitzer. The random discrete action for two-dimensional spacetime. Classical and Quantum Gravity, 2011, 28 (10), pp.105018. ⟨10.1088/0264-9381/28/10/105018⟩. ⟨hal-00699374⟩